Single-Particle Electrodynamics - Assassination Science

(6.99) then gives
{
(a·b) − 5(n·a)(n·b) r2
˜r 2 } 3
4π˜r 3 {
1 − r2
˜r 2 }
.
We may now trivially integrate over the angular coördinates n of r—since
n involves r only, having no reference to the appended coördinate w—giving
(a·b)/3. We thus complete the integration over r:
∫ { }
∞
(a·b) 4πr 2 3
dr 1 − r2
− 5 ∫ ∞
0 4π˜r 3 ˜r 2 3 (a·b) 4πr 2 dr 3r2
0
again employing the change of variable (6.87), we find
∫ 1
∫ 1
3(a·b) du u 2 − 5(a·b) du u 4 = 0.
0
0
Thus, we find that the divergence (6.99) in fact vanishes:
∇·
4π˜r 5 {
1 − r2
˜r 2 }
;
(a·b)n − 3(n·a)(n·b)n
4πr 2 = 0, (6.100)
where again we drop the tildes in this final expression.
We have thus shown that there are no further delta-function contributions
to the divergences of the point field expressions. The vanishing of the
divergences of the regular function terms is demonstrated by the computer
algebra program radreact of Section G.6.
6.7 Inverse-cube integrals
In the considerations of Section 6.4, we generally assumed that the angular
integration over n d , and radial integration over r d , may be computed separately.
We noted there, however, that we must take special care when it
comes to inverse-cube integrals that we shall have need to perform, viz., those
of the form
η 0
( 4
3 πε3 ) −2 ∫
∫
d 3 r
r≤ε
274
d 3 r ′ rd −3 f(n d), (6.101)
r ′ ≤ε